graph LR
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    idea_0d2de2b4_45c2_4493_b236_c14023abd04a["Dot Products and Duality: Vectors as Linear Transformations"]
    idea_0d9d721b_bde2_4360_95f1_b76d81ae29b9["Dual Vectors in Three-Dimensional Space"]
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    idea_479f0cd6_f0cc_4263_894b_88487029642a["Linear Transformations of 3D Vectors via Matrix Representations"]
    idea_4e88b2fb_27db_442e_957e_cec98cb9a6ae["Signed Volume Calculations for Parallelepipeds"]
    idea_7b1b274b_79f5_4148_aa7e_f8ac6e880fe8["Linear Independence in Vector Spaces"]
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    idea_a7ab481a_2105_4be2_b5e7_c8396fc49352["Mathematics for nonmathematicians (engineering, social sciences, etc.)"]
    idea_db76f33e_5ec7_46b5_be03_a759f24998e1["Solving Systems of Linear Equations Algebraically"]
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